# Catalan's Conjecture

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## Theorem

The only solution to the Diophantine equation:

- $x^a - y^b = 1$

for $a, b > 1$ and $x, y > 0$, is:

- $x = 3, a = 2, y = 2, b = 3$

## Proof

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## Also known as

**Catalan's Conjecture** is now also known as Mihăilescu's Theorem, for Preda V. Mihăilescu who proved it true.

## Also see

- Consecutive Integers which are Powers of 2 or 3: the special case where $x$ and $y$ are $2$ and $3$
- 1 plus Power of 2 is not Perfect Power except 9: the special case of $y = 2$.
- 1 plus Perfect Power is not Power of 2: the special case of $x = 2$.
- 1 plus Square is not Perfect Power: the special case of $b = 2$.
- 1 plus Perfect Power is not Prime Power except for 9: the special case where $x$ is prime.

## Source of Name

This entry was named for Eugène Charles Catalan.

## Historical Note

Catalan's Conjecture was first put forward by Eugène Charles Catalan in $1844$.

The case where $x$ and $y$ are $2$ and $3$ was proved in $1344$ by Levi ben Gershon.

A proof was finally published in $2004$ by Preda V. Mihăilescu.

## Sources

- 1944: Eugène Catalan:
*Note extraite d'une lettre adressée à l'éditeur*(*J. reine angew. Math.***Vol. 27**: p. 192) - 2004: Preda Mihăilescu:
*Primary Cyclotomic Units and a Proof of Catalan's Conjecture*(*J. reine angew. Math.***Vol. 572**: pp. 167 – 195) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Catalan's conjecture**